An important observation in Refs. [1]}, [2]} was that the damping term in Eq. (REF ) renders the description of gauge-field production during axion inflation inherently nonlocal in time.
Gauge-field modes inside the Hubble horizon experience an exponential damping on their approach to horizon crossing because they no longer evolve in an empty de Sitter vacuum but in a conducting medium.
The amount of exponential damping, however, depends on the conductivity of the fermion gas, which in turn is sensitive to the efficiency of gauge-field and fermion production at earlier times.
This effect can be captured by a new damping factor\(7\endcsname \)The lower integration boundary in Eq. (REF ) implies that gauge-field modes on arbitrarily small scales inside the horizon will experience damping because of the nonvanishing conductivity.
This corresponds to a technical simplification that does not necessarily reflect the actual physical situation:
Gauge-field modes on scales smaller than the typical momenta in the fermion gas are expected to experience less damping up to no damping at all.
The integral in Eq. (REF ) is, however, typically dominated by the contributions at late times, i.e., contributions close to the upper integration boundary.
Our simplified treatment of the lower integration boundary is therefore expected to have no or only little phenomenological implications.
See also Ref. [2]} for a more comprehensive discussion.

\(\Delta \left(\tau \right) = \exp \left[- \int _{-\infty }^\tau d\tau ^{\prime } \sigma \left(\tau ^{\prime }\right)\right] \,,\) 
